Question: What is the value of $\dfrac{d}{dx}\left(x^{-7}\right)$ at $x=-1$ ?
Explanation: Let's first find the expression for $\dfrac{d}{dx}\left(x^{-7}\right)$ and then evaluate it at $x=-1$. The derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a negative number.) $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{-7}\right) \\\\ &=-7x^{-7-1} \gray{\text{The power rule}} \\\\ &=-7x^{-8} \end{aligned}$ So we found that $\dfrac{d}{dx}\left(x^{-7}\right)=-7x^{-8}$, which can also be written as $-\dfrac{7}{x^8}$. Now let's plug ${x=-1}$ : $\begin{aligned} -\dfrac{7}{({-1})^8}&=-\dfrac{7}{1} \\\\ &=-7 \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\left(x^{-7}\right)$ at $x=-1$ is $-7$.